Inequality in $\mathbb{C}$ using continuous functions

Question

Let $f:[a, b] \rightarrow \mathbb{C}$ be a continuous and differentiable function (which means that both $Re(f)$ and $Im(f)$ are differentiable and $f' = Re(f)' + iIm(f)'$). We are to proof that:
$$|f(b) - f(a)| \le |b-a|\sup_{t \in [a,b]}|f'(t)|$$ I tried to solve that problem using Lagrange theorem but unfortunately I failed. I would appreciate any tips or hints.

Answer 1

There exists a complex number $\alpha$ with $|\alpha|=1$ such that $\alpha(f(b)-f(a))\ge0$. Define $g(t)=\Re(\alpha f(t))$ and apply the Mean Value Theorem to $g$; then use the fact that $|g'|\le|f'|$.